A question about clopen set From wikipedia: "Now consider the space X which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set (0,1) is clopen, as is the set (2,3)." why (0,1) is clopen?

The set $(0,1)$ is open as a subset of $\mathbb{R}$ and therefore also open in the subspace topology induced on $(0,1) \cup (2,3)$. The same holds for $(2,3)$. Since the complement of $(0,1)$ in $(0,1) \cup (2,3)$ is $(2,3)$ and therewith open, $(0,1)$ is also closed in $(0,1) \cup (2,3)$. Hence it is clopen.